Let \(R\) be a commutative ring with identity and let \(M\) be an \(R\)-module. The principle of idealization of \(R\) and \(M\), due to Nagata, is the ring \(R(M):=R\oplus M\) with addition \((a,x)+ (b,y)= (a+b, x+y)\) and multiplication \((a,x)(b,y)= (ab, ay+bx)\), for all \(a,b\in R\) and \(x,y\in M\). It is known that \(R(M)\) is a commutative ring with identity. Moreover, \(R(M)\) is Noetherian if and only if \(R\) is Noetherian and \(M\) is finitely generated.
For an integral domain \(D\), let \(M_2(D)\) denote the ring of \(2\times 2\) matrices over \(D\). Set \[ R(D):=\{\left( \begin{matrix} a & b \\ 0 & a \\ \end{matrix} \right)|\, a,b \in D\}. \] Then \(R(D)\) is a subring of \(M_2(D)\) and the map \(a\longmapsto \left( \begin{matrix} a & 0 \\ 0 & a \\ \end{matrix} \right)\) is a monomorphism of \(D\) into \(R(D)\). Moreover, the map \(\phi:R(M)\longrightarrow R(D)\), defined by \(\phi(a, b)=\left( \begin{matrix} a & b \\ 0 & a \\ \end{matrix} \right)\) is a ring isomorphism. In view of this isomorphism, \(R(D)\) is called self-idealization of \(D\). If \(D\) is a principal ideal domain (PID), then \(R(D)\) is a Noetherian ring and so \(R(D)\) is atomic (i.e. every non-zero and non-unit element of \(R(D)\) can be written as a finite product of irreducible elements).
In this paper under review the authors completely characterize the irreducible elements of \(R(D)\) and then they use this result to show how to factorize each non-zero and non-unit elements of \(R(D)\) into irreducible elements via the factorization of \(D\). More precisely, they show that an element \(\left( \begin{matrix} a & b \\ 0 & a \\ \end{matrix} \right)\) of \(R(D)\) is irreducible if and only if either, (i) \(a=0\) and \(b\in D\) is unit, (ii) \(a=p\) or (iii) \(a=up^n\) and \(1\in GCD(a, b)\) for some prime \(p\in D\) and integer \(n\geq2\), where \(u\) is an unit in \(D\).
Also, the authors describe the system of sets of lengths \(\mathcal{L}(R(D))\). Recall that the system of sets of lengths of a Noetherian ring \(R\) is defined as \[ \mathcal{L}(R(R)):=\{L(x)|\, x\in R\setminus \{0\} \}, \] where \[ L(x)=\{n\in \mathbb{N}|\, \text{there exist irreducible elements}\, u_1, \dots, u_n \text{with}\,\, x=u_1\cdot \dots\cdot u_n\}. \]